State Evolution for Approximate Message Passing with Non-Separable Functions

Given a high-dimensional data matrix ${\boldsymbol A}\in{\mathbb R}^{m\times n}$, Approximate Message Passing (AMP) algorithms construct sequences of vectors ${\mathit{u}}^t\in {\mathbb{R}}^n$, ${\boldsymbol v}^t\in{\mathbb R}^m$,indexedby$t\in{0,1,2\dots}$byiterativelyapplying${\boldsymbol A}$or${\boldsymbol A}^{{\sf T}}$, and suitable non-linear functions, which depend on the specific application. Special instances of this approach have been developed --among other applications-- for compressed sensing reconstruction, robust regression, Bayesian estimation, low-rank matrix recovery, phase retrieval, and community detection in graphs. For certain classes of random matrices $\mathit{A}$, AMP admits an asymptotically exact description in the high-dimensional limit $m,n\to \mathrm{\infty }$, which goes under the name of `state evolution.’ Earlier work established state evolution for separable non-linearities (under certain regularity conditions). Nevertheless, empirical work demonstrated several important applications that require non-separable functions. In this paper we generalize state evolution to Lipschitz continuous non-separable nonlinearities, for Gaussian matrices $\mathit{A}$. Our proof makes use of Bolthausen’s conditioning technique along with several approximation arguments. In particular, we introduce a modified algorithm (called LAMP for Long AMP) which is of independent interest.